|6 March (Wednesday)
|Department of Mathematics and Statistics
McGill University, Toronto
|This is the second lecture of Professor Makkai's lectures series on FOLDS:
Lecture I 1 March, 4:15 PM, Room 226 (LaPoM)
Lecture II 6 March, 5:00 PM, Room 226 (TPF)
Lecture III 8 March, 4:15 PM, Room 226 (LaPoM)
|Lectures on FOLDS II.
FOLDS, first-order logic with
dependent sorts, was introduced by me in a 1995 monograph
(www.math.mcgill.ca/makkai/). It is explained in outline in the paper
"Towards a categorical foundation of mathematics" published in Springer
Lecture Notes in Logic, no.11, 1998. FOLDS is intended as the formal
language for a foundational system based on higher dimensional
categories, analogously to ordinary first-order logic in ZFC set theory.
The syntax of FOLDS is simple; it is related, although not identical,
to earlier formalisms by Per Martin-Lof and John Cartmell. However,
unlike the latter, FOLDS is a fully model-theoretical language, with a
general Tarskian semantics and an accompanying model theory. Its
main feature is a replacement of ordinary (Fregean) equality (as in
"logic with equality") by a new, signature-dependent notion of FOLDS
equivalence. In its most direct application, FOLDS equivalence takes the
role of isomorphism of ordinary model theory; but it also specializes,
in the appropriate contexts, to equivalence of categories, biequivalence
of bicategories, etc. The most important application of FOLDS
equivalences appear in my definition of a universe called "The
multitopic category of all multitopic categories" (1999 and 2004; the
In the lectures, I will try to
describe the concepts and their theory through examples, rather than
general formulations. The project of the new "categorical" foundation
has as its aim the establishing of a self-contained formal theory of
totalities that is workable, and also fundamentally different from
ordinary set-theory insofar it should be free from smallness
considerations of the Cantorian type. This project is far from being
completed, and the audience is invited to join in the investigation of
the several precise mathematical questions as well as the more general
philosophical aspects of the project.
|13 March (Wednesday)
|Department of Modern Philosophy, Institute of Philosophy Eötvös University, Budapest|
|A filozófia mint praxis
(Philosophy as practice)
európai filozófia története a filozófia tudományossá és elméletivé
válásának története. Ennek természetesen megvannak a pozitív eredményei,
de vannak negatívumai is. Ez utóbbiak közül talán a legfontosabb az,
hogy az antikvitásban még magától értetődő praktikus és terápiás jelleg
eltűnt a filozófiai hagyományból. A filozófiai gyakorlatot felváltotta a
gyakorlati filozófia (etika, morálfilozófia, axiológia,
cselekvéselmélet), a terápiás jelleg pedig elméleti kritikává vált. A
XX. században megjelentek olyan irányzatok, amelyek arra tettek
kísérletet, hogy a filozófiát ismét gyakorlattá, filozófiai
lélekgyakorlattá tegyék, vagyis a terápia fogalmát mintegy
visszahódítsák a pszichológiától. Az előadás ennek a tendenciának a
hátterét, lehetőségeit és jövőjét igyekszik tisztázni.
|20 March (Wednesday)
|Department of Logic, Institute of Philosophy|
Eötvös University, Budapest
|Mass and Modality
Logic and Theory of Relativity group lead by Andréka, H. and Németi, I.
developed several axiom systems for relativity theory to investigate it
within mathematical logic.
One of the simplest and most commonly used axiom system is an axiom
system of kinematics, the so-called SpecRel. Although this axiom system
is very simple, it implies all the main predictions (theorems) of
special relativity theory. However, as it is proposed by the group in
many articles, sometimes the classical first-order logic framework of
SpecRel does not seem to be sufficient to give back the appropriate
physical meaning. For example, the main axiom of SpecRel, the axiom
which is about the possibility of sending out light signals, states that
there could be a photon which crosses certain points. This "could be"
indicates some kind of notion of possibility, which is barely accessible
from a classical first-order logic.
This problem becomes more serious when we try to expand the system
SpecRel by certain dynamical axioms (to get SpecRelDyn). For example, we
would like to postulate that for every observer, everywhere, any kind
of possible collision is realizable. It is worth to investigate this
type of axioms, because this way leads to an experimental understanding
of the notion of possibility.
We will investigate axiom systems of special relativity based on modal
logic, which is the standard tool for formally handle dynamical notions –
such as performing a (thought-) experiment, for instance "send out a
light signal" or "realize a collision".
Our axiom systems will be built with the following goals:
- Give a plausible but formal notion of possibility/experimentation
based on the informal explanations of the classical SpecRel and
- Save the theorems and the ideas of their proofs from SpecRel and SpecRelDyn.
- Show that in a modal framework the mass can be explicitly defined
essentially in the language of kinematics. This can be viewed as the
formal interpretation of the operational definition of mass.